Computer-assisted existence proofs for one-dimensional Schrödinger-poisson systems
Motivated by the three-dimensional time-dependent Schr¨odinger-Poisson system we prove the existence of non-trivial solutions of the one-dimensional stationary Schr¨odinger-Poisson system using computer-assisted methods. Starting from a numerical approximate solution, we compute a bound for its defe...
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| Dokumentumtípus: | Cikk |
| Megjelent: |
University of Szeged, Institute of Informatics
Szeged
2020
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| Sorozat: | Acta cybernetica
24 No. 3 |
| Kulcsszavak: | Számítástechnika, Kibernetika |
| Tárgyszavak: | |
| doi: | 10.14232/actacyb.24.3.2020.6 |
| Online Access: | http://acta.bibl.u-szeged.hu/69282 |
| Tartalmi kivonat: | Motivated by the three-dimensional time-dependent Schr¨odinger-Poisson system we prove the existence of non-trivial solutions of the one-dimensional stationary Schr¨odinger-Poisson system using computer-assisted methods. Starting from a numerical approximate solution, we compute a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution. For the latter, eigenvalue bounds play a crucial role, especially for the eigenvalues “close to” zero. Therefor, we use the Rayleigh-Ritz method and a corollary of the Temple-Lehmann Theorem to get enclosures of the crucial eigenvalues of the linearization below the essential spectrum. With these data in hand, we can use a fixed-point argument to obtain the desired existence of a non-trivial solution “nearby” the approximate one. In addition to the pure existence result, the used methods also provide an enclosure of the exact solution. |
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| Terjedelem/Fizikai jellemzők: | 373-391 |
| ISSN: | 0324-721X |